On Well Quasi Orders of Free Monoids

The paper investigates down-sets associated to well quasi orders. Of particular language-theoretic interest is the quasi order u s v (resp.u P v) of u being a subword (resp. a Parikh subword) of v, as well as their inverses. We establish a number of results about the regularity and e ective regularity of the down-sets, in particular, a general condition for the e ective regularity of the down-set associated to an inverse well quasi order (Theorem 7.1). It is decidable whether or not an arbitrary regular language results as a down-set from an in nite or bi-in nite word, whereas the same problem is undecidable for context-free language. A quasi order being a well quasi order is connected to arbitrary languages being con uent.